This paper is meant for the newbie. it's not a country­ of-the-art paper for examine staff within the box of regulate idea. Its goal is to introduce the reader to a couple of the issues and ends up in keep watch over conception, to demonstrate the appliance of those re­ sults, and to supply a advisor for his extra interpreting in this topic. i've got attempted to encourage the consequences with examples, especial­ ly with one canonical, basic instance defined in §3. Many effects, akin to the utmost precept, have lengthy and tough proofs. i've got passed over those proofs. regularly i've got integrated in simple terms the proofs that are both (1) now not too tough or (2) really enlightening as to the character of the end result. i've got, although, frequently tried to attract the most powerful end from a given facts. for instance, many present proofs up to the mark concept for compact goals and forte of options additionally carry for closed objectives and non-uniqueness. eventually, on the finish of every part i've got given references to generalizations and origins of the consequences mentioned in that part. I make no declare of completeness within the references, notwithstanding, as i've got usually been content material simply to refer the reader both to an exposition or to a paper which has an in depth bibliography. IV those 1ecture notes are revisions of notes I used for aseries of 9 1ectures on contro1 concept on the foreign summer time Schoo1 on Mathematica1 structures and Economics held in Varenna, Ita1y, June 1967.

CONTROLLABILITY USING SPECIAL CONTROLS We now consider the same problems as those in §4 for certain U of UM' subclasses classes In particular, we want to know for which U properly contained in unchanged. the controllable set ~ K remains We shall show that in certain cases the classes of piece- wise constant controls and bang-bang controls are two such classes. Throughout this section we consider only the zero target o G(t) for Piecewise constant controls. is piecewise constant if [t o ' t l ] intervals, on each of which of such u in sideration, let ~ u(') for some ~C is the union of finitely many is constant.

0 J so that Since K (t ) o the ball of radius M/2 true for every K = Rn or K (t ) 0 . u. < 0 J Let M> 0 contains the d th axis d was arbitrary, there exists for each Thus t o , But . x E,K- (t ) o 0 contains every axis from is now convex, so is M, , component of some < 0 0 00 so that either u* is the u* i-d 0 Hence we can force either of the relations 00. ~ every (t -s) - M to hence M. Since K-(t) o K contains this ball. 2) contains Since this is The proof is now complete. in the linear system form (LA), 0 ~ We now apply these results to the railroad train example.